Estimates on conjectures of Minkowski and woods

Hans-Gill, R. J. ; Raka, Madhu ; Sehmi, Ranjeet (2010) Estimates on conjectures of Minkowski and woods Indian Journal of Pure and Applied Mathematics, 41 (4). pp. 595-606. ISSN 0019-5588

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Let Rn be the n-dimensional Euclidean space. Let Λ be a lattice of determinant 1 such that there is a sphere |X| < R which contains no point of Λ other than the origin O and has n linearly independent points of Λ on its boundary. A well known conjecture in the geometry of numbers asserts that any closed sphere in Rn of radius √n/4 contains a point of Λ. This is known to be true for n ≤ 8. Here we give estimates on a more general conjecture of Woods for n ≥ 9. This leads to an improvement for 9 ≤ n ≤ 22 on estimates of Il'in (1991) to the long standing conjecture of Minkowski on product of n non-homogeneous linear forms.

Item Type:Article
Source:Copyright of this article belongs to Indian National Science Academy.
Keywords:Lattice; Covering; Non-homogeneous; Product of linear forms; Critical determinant; Korkine and Zolotareff reduction; Hermite's constant; Centre density
ID Code:12285
Deposited On:10 Nov 2010 05:16
Last Modified:03 Jun 2011 06:26

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